Theorem relcmpcmet 25285

Description: If 𝐷 is a metric space such that all the balls of some fixed size are relatively compact, then 𝐷 is complete. (Contributed by Mario Carneiro, 15-Oct-2015.)

Hypotheses

Ref	Expression
relcmpcmet.1	⊢ 𝐽 = (MetOpen‘𝐷)
relcmpcmet.2	⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
relcmpcmet.3	⊢ (𝜑 → 𝑅 ∈ ℝ+)
relcmpcmet.4	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp)

Assertion

Ref	Expression
relcmpcmet	⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))

Distinct variable groups:   𝑥,𝐷   𝑥,𝐽   𝜑,𝑥   𝑥,𝑅   𝑥,𝑋

Proof of Theorem relcmpcmet

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		relcmpcmet.2	. 2 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
2		metxmet 24299	. . . . . . 7 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
3	1, 2	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
4	3	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) → 𝐷 ∈ (∞Met‘𝑋))
5		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) → 𝑓 ∈ (CauFil‘𝐷))
6		relcmpcmet.3	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ ℝ+)
7	6	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) → 𝑅 ∈ ℝ+)
8		cfil3i 25236	. . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑓 ∈ (CauFil‘𝐷) ∧ 𝑅 ∈ ℝ+) → ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)
9	4, 5, 7, 8	syl3anc 1374	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) → ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)
10	3	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝐷 ∈ (∞Met‘𝑋))
11		relcmpcmet.1	. . . . . . . . 9 ⊢ 𝐽 = (MetOpen‘𝐷)
12	11	mopntopon 24404	. . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
13	10, 12	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝐽 ∈ (TopOn‘𝑋))
14		cfilfil 25234	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑓 ∈ (CauFil‘𝐷)) → 𝑓 ∈ (Fil‘𝑋))
15	3, 14	sylan 581	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) → 𝑓 ∈ (Fil‘𝑋))
16	15	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑓 ∈ (Fil‘𝑋))
17		simprr 773	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)
18		topontop 22878	. . . . . . . . . . 11 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
19	13, 18	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝐽 ∈ Top)
20		simprl 771	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑥 ∈ 𝑋)
21	6	rpxrd 12987	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ ℝ*)
22	21	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑅 ∈ ℝ*)
23		blssm 24383	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑥(ball‘𝐷)𝑅) ⊆ 𝑋)
24	10, 20, 22, 23	syl3anc 1374	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ⊆ 𝑋)
25		toponuni 22879	. . . . . . . . . . . 12 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
26	13, 25	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑋 = ∪ 𝐽)
27	24, 26	sseqtrd 3958	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ⊆ ∪ 𝐽)
28		eqid 2736	. . . . . . . . . . 11 ⊢ ∪ 𝐽 = ∪ 𝐽
29	28	clsss3 23024	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (𝑥(ball‘𝐷)𝑅) ⊆ ∪ 𝐽) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ ∪ 𝐽)
30	19, 27, 29	syl2anc 585	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ ∪ 𝐽)
31	30, 26	sseqtrrd 3959	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋)
32	28	sscls 23021	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ (𝑥(ball‘𝐷)𝑅) ⊆ ∪ 𝐽) → (𝑥(ball‘𝐷)𝑅) ⊆ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))
33	19, 27, 32	syl2anc 585	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ⊆ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))
34		filss 23818	. . . . . . . 8 ⊢ ((𝑓 ∈ (Fil‘𝑋) ∧ ((𝑥(ball‘𝐷)𝑅) ∈ 𝑓 ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ⊆ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓)
35	16, 17, 31, 33, 34	syl13anc 1375	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓)
36		fclsrest 23989	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓) → ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) = ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
37	13, 16, 35, 36	syl3anc 1374	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) = ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
38		inss1 4177	. . . . . . 7 ⊢ ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ⊆ (𝐽 fClus 𝑓)
39		eqid 2736	. . . . . . . . 9 ⊢ dom dom 𝐷 = dom dom 𝐷
40	11, 39	cfilfcls 25241	. . . . . . . 8 ⊢ (𝑓 ∈ (CauFil‘𝐷) → (𝐽 fClus 𝑓) = (𝐽 fLim 𝑓))
41	40	ad2antlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽 fClus 𝑓) = (𝐽 fLim 𝑓))
42	38, 41	sseqtrid 3964	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ⊆ (𝐽 fLim 𝑓))
43	37, 42	eqsstrd 3956	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ⊆ (𝐽 fLim 𝑓))
44		relcmpcmet.4	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp)
45	44	ad2ant2r 748	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp)
46		filfbas 23813	. . . . . . . . . 10 ⊢ (𝑓 ∈ (Fil‘𝑋) → 𝑓 ∈ (fBas‘𝑋))
47	16, 46	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑓 ∈ (fBas‘𝑋))
48		fbncp 23804	. . . . . . . . 9 ⊢ ((𝑓 ∈ (fBas‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓) → ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓)
49	47, 35, 48	syl2anc 585	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓)
50		trfil3 23853	. . . . . . . . 9 ⊢ ((𝑓 ∈ (Fil‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋) → ((𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ↔ ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓))
51	16, 31, 50	syl2anc 585	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ↔ ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓))
52	49, 51	mpbird 257	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
53		resttopon 23126	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋) → (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (TopOn‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
54	13, 31, 53	syl2anc 585	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (TopOn‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
55		toponuni 22879	. . . . . . . . 9 ⊢ ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (TopOn‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) = ∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
56	54, 55	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) = ∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
57	56	fveq2d 6844	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) = (Fil‘∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))))
58	52, 57	eleqtrd 2838	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))))
59		eqid 2736	. . . . . . 7 ⊢ ∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) = ∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))
60	59	fclscmpi 23994	. . . . . 6 ⊢ (((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp ∧ (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘∪ (𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))) → ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ≠ ∅)
61	45, 58, 60	syl2anc 585	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ≠ ∅)
62		ssn0 4344	. . . . 5 ⊢ ((((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ⊆ (𝐽 fLim 𝑓) ∧ ((𝐽 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓 ↾t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ≠ ∅) → (𝐽 fLim 𝑓) ≠ ∅)
63	43, 61, 62	syl2anc 585	. . . 4 ⊢ (((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥 ∈ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽 fLim 𝑓) ≠ ∅)
64	9, 63	rexlimddv 3144	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (CauFil‘𝐷)) → (𝐽 fLim 𝑓) ≠ ∅)
65	64	ralrimiva 3129	. 2 ⊢ (𝜑 → ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅)
66	11	iscmet 25251	. 2 ⊢ (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅))
67	1, 65, 66	sylanbrc 584	1 ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  ∃wrex 3061   ∖ cdif 3886   ∩ cin 3888   ⊆ wss 3889  ∅c0 4273  ∪ cuni 4850  dom cdm 5631  ‘cfv 6498  (class class class)co 7367  ℝ^*cxr 11178  ℝ⁺crp 12942   ↾_t crest 17383  ∞Metcxmet 21337  Metcmet 21338  ballcbl 21339  fBascfbas 21340  MetOpencmopn 21342  Topctop 22858  TopOnctopon 22875  clsccl 22983  Compccmp 23351  Filcfil 23810   fLim cflim 23899   fClus cfcls 23901  CauFilccfil 25219  CMetccmet 25221

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-fi 9324  df-sup 9355  df-inf 9356  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-n0 12438  df-z 12525  df-uz 12789  df-q 12899  df-rp 12943  df-xneg 13063  df-xadd 13064  df-xmul 13065  df-ico 13304  df-rest 17385  df-topgen 17406  df-psmet 21344  df-xmet 21345  df-met 21346  df-bl 21347  df-mopn 21348  df-fbas 21349  df-fg 21350  df-top 22859  df-topon 22876  df-bases 22911  df-cld 22984  df-ntr 22985  df-cls 22986  df-nei 23063  df-cmp 23352  df-fil 23811  df-flim 23904  df-fcls 23906  df-cfil 25222  df-cmet 25224

This theorem is referenced by:  cmpcmet  25286  cncmet  25289